Rigidity and a mesoscopic central limit theorem for Dyson Brownian motion for general $$\beta $$ β and potentials

Abstract

We study Dyson Brownian motion with general potential V and for general $$\beta \ge 1$$ . For short times $$t = o (1)$$ and under suitable conditions on V we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale $$\eta _* \ll t \ll 1$$ , we prove a mesoscopic central limit theorem for linear statistics at all scales $$\eta $$ with $$N^{-1}\ll \eta \ll t$$ .